Measure doubling in unimodular locally compact groups and quotients

Abstract

We consider a (possibly discrete) unimodular locally compact group $G$ with Haar measure ${\mu }_G$, and a compact $A\subseteq G$ of positive measure with ${\mu }_G(A^2)\le K{\mu }_G(A)$. Let $H$ be a closed normal subgroup of G and $\pi :G\to G/H$ be the quotient map. With the further assumption that $A=A^{-1}$, we show $${\mu }_{G/H}(\pi A^2)\le K^2{\mu }_{G/H}(\pi A).$$ We also demonstrate that $K^2$ cannot be replaced by $(1-\epsilon )K^2$ for any $\epsilon >0$. In the general case (without $A=A^{-1}$), we show ${\mu }_{G/H}(\pi A^2)\le K^3{\mu }_{G/H}(\pi A)$, improving an earlier result by An, Jing, Zhang, and the third author. Moreover, we are able to extract a compact set $B\subseteq A$ with ${\mu }_G(B)>{\mu }_G(A)/2$ such that ${\mu }_{G/H}(\pi B^2)<2K{\mu }_{G/H}(\pi B)$.

12 pages